If you have good input/output isolation (i.e. cascode) then you could
1. proceed with the NFmin sweep versus bias
2. find the bias current for minimum NF
3. conjugate match the output at the best bias current for lowest NF
4. repeat step 1 with the output conjugate match in the circuit

This approach should give you an idea of the delta gain difference between gain at NFmin with and without output conjugate matching.

If there is poor input/output isolation the procedure doesn't work well because the output and input impedances are interdependent on each other.

I do not clearly understand your second question. Cascode amplifier has several differences compared to a common source amplifier. The most notable differences are good input/output isolation at high frequencies, lower input capacitance (minimizing Miller effect), and higher output impedance. Yes, there is a penalty in voltage head room to keep the device in saturation, but the impact depends on the available supply voltage and the linearity performance requirements.

A shunt output inductor to the supply improves the available output voltage swing range and possibly gain (resonating out capacitance). But you do not have to use an inductor. It all depends your requirements.